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Evaluating an infinite series Write the Maclaurin series for f(x) = ln (1+x) and find the interval of convergence. Evaluate f(−1/2) to find the value of ∑ₖ₌₁∞ 1/(k 2ᵏ)
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 11.2.51
Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
g(x) = 2/(1 − 2x)² using f(x) = 1/(1 − 2x)
Guida verificata passo dopo passo1
Recall the given function f(x) = \(\frac{1}{1 - 2x}\), which can be expressed as a geometric series centered at 0. The general form of a geometric series is \(\sum\)_{n=0}^\(\infty\) r^n = \(\frac{1}{1-r}\) for |r| < 1. Here, r = 2x, so the power series for f(x) is \(\sum\)_{n=0}^\(\infty\) (2x)^n = \(\sum\)_{n=0}^\(\infty\) 2^n x^n, valid for |2x| < 1 or |x| < \(\frac{1}{2}\).
Notice that g(x) = \(\frac{2}{(1 - 2x)^2}\) is related to f(x) by differentiation. Specifically, the derivative of f(x) with respect to x is f'(x) = \(\frac{d}{dx}\) \(\left\)( \(\frac{1}{1 - 2x}\) \(\right\)). Use the chain rule to find f'(x).
Calculate f'(x): Since f(x) = (1 - 2x)^{-1}, then f'(x) = -1 \(\cdot\) (1 - 2x)^{-2} \(\cdot\) (-2) = \(\frac{2}{(1 - 2x)^2}\). This shows that g(x) = f'(x).
Differentiate the power series for f(x) term-by-term to find the power series for g(x). The term-by-term differentiation of \(\sum\)_{n=0}^\(\infty\) 2^n x^n is \(\sum\)_{n=1}^\(\infty\) n 2^n x^{n-1}. This series represents g(x).
Determine the interval of convergence for g(x). Since differentiation does not change the radius of convergence, the interval of convergence for g(x) is the same as for f(x), which is |x| < \(\frac{1}{2}\).
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Power Series Representation
A power series expresses a function as an infinite sum of terms involving powers of the variable, typically centered at a point (here, 0). Understanding how to write functions like f(x) = 1/(1 − 2x) as a power series is essential, as it forms the basis for manipulating and finding related series for functions like g(x).
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05:58
Intro to Power Series
Term-by-Term Differentiation and Integration of Power Series
Power series can be differentiated or integrated term-by-term within their interval of convergence. This property allows us to find the power series for g(x) by differentiating or integrating the known series for f(x), enabling the construction of new series representations from existing ones.
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05:58Intro to Power Series
Interval of Convergence
The interval of convergence is the set of x-values for which a power series converges. When differentiating or integrating a power series, the interval of convergence may change or remain the same, so determining this interval for the resulting series is crucial to ensure the series accurately represents the function.
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08:44Interval of Convergence
Pratica correlata
Domanda del libro di testo
Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.
cos 2
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Use of Tech Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at a.
c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
Find the Taylor polynomials p₁, p₂, and p₃ centered at a=1 for f(x)=x³.
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Domanda del libro di testo
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k
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Domanda del libro di testo
Combining power series Use the geometric series
f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,
to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.
g(x) = x³/(1 − x)
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Domanda del libro di testo
Radius of convergence Find the radius of convergence for the following power series.
∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)
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